STATISTICAL CONSIDERATIONS

In many areas of research where controlled experiments are not practicable, the techniques of multiple regression analysis are commonly used to quantify the effects of different X-variables on some response Y. The determination of the relationship of a dependent variable Y, with a number of independent variables Y^ is commonly required in forestry research, and is the problem encountered in this study.

Specific reference to the use of linear regression analysis techniques in forestry research is made in Freese [1964], Snedecor and Cochran [1967], Draper and Smith [1966] and Leech [1973]. Linear analysis refers to linearity in the parameters of the type

Y = + + $2^2 ••• ^p^p + ^ *

X^, X ^ j X ^ . ^ o X ^ ; 3q ,^^ e t c . r e p r e s e n t c o e f f i c i e n t s (w hich a r e c o n s t a n t s ) and E i s an e r r o r t e r m . N o n - l i n e a r r e g r e s s i o n a n a l y s i s r e f e r s t o f u n c t i o n s which a r e n o n ­ l i n e a r i n t h e p a r a m e t e r s , f o r exam ple Y =

91

(VV - e-Vl ♦ e.

8 1 - 02 L J N o n - l i n e a r r e g r e s s i o n a n a l y s e s a r e l e s s commonly u s e d a s t h e y he e n o t b e e n i n t e g r a t e d i n t o a t h e o r y o f t h e same d e g r e e o f g e n e r a l i t y a s t h a t w hich a p p l i e s t o t h e l i n e a r c a s e ( S n e d e c o r and C o c h ra n , 1967; F r e e s e , 1 9 6 4 ) . I n t h i s s t u d y , an a p p ro a c h u s i n g l i n e a r r e g r e s s i o n a n a l y s i s was a d o p t e d f o r t h r e e main r e a s o n s : 1. A lth o u g h c o n s i d e r a b l e ti m e had b e e n s p e n t i n t h e f i e l d , t h e amount and r a n g e o f t h e d a t a o b t a i n e d was l i m i t e d b y b o th t h e t i m e a v a i l a b l e f o r c o l l e c t i o n and t h e s i t e s and t r u c k s a v a i l a b l e f o r o b s e r v a t i o n . More r e f i n e d a n a l y s e s would n o t t h e r e f o r e im p ro v e t h e r e s u l t s o b t a i n e d .

2. The form o f t h e t r u e model was n o t a v a i l a b l e b y s im p le t h e o r e t i c a l c o n s i d e r a t i o n s and t h e r e f o r e t h e r e was no i n d i c a t i o n t h a t no n ­

l i n e a r form s would a c t u a l l y be r e q u i r e d .

3 . P a c k a g e co m p u te r p ro g ram s w hich c o u l d b e i n d e p e n d e n t l y h a n d le d and i n t e r p r e t e d were r e a d i l y a v a i l a b l e f o r l i n e a r r e g r e s s i o n , and i n p a r t i c u l a r , t h e REX p ro g ram (G ro se n b a u g h , 1 9 6 7 ).

A number o f t h e more g e n e r a l p r o b le m s i n m u l t i p l e l i n e a r r e g r e s s i o n a n a l y s i s w ere a p p l i c a b l e t o t h i s s t u d y . In t h i s t y p e o f a n a l y s i s , t h e r e s p o n s e o f t h e d e p e n d e n t v a r i a b l e (Y) i s gauged a g a i n s t t h e e f f e c t s o f d i f f e r e n t in d e p e n d e n t v a r i a b l e s (X’ s ) . I t was n o t p o s s i b l e i n t h e t y p e o f m easu re m en ts made i n t h i s s t u d y t o e n s u r e t h a t X v a l u e s o t h e r t h a n p o w e r , w e ig h t and g r a d e were n o t r e l a t e d t o Y ( s p e e d ) i n t h e s am p led p o p u l a t i o n . These o t h e r v a l u e s a r e d i s c u s s e d i n S e c t i o n 3 . 4 . 1 . I f one o r more o t h e r r e l a t e d X v a l u e s e x i s t , t h e e s t i m a t e d v a l u e o f t h e c o e f f i c i e n t s o f X i n t h e computed r e g r e s s i o n from t h e sa m p le would n o t b e u n b i a s s e d e s t i m a t e s o f t h e p o p u l a t i o n ’ s c o e f f i c i e n t s ( S n e d e c o r and C o c h ra n , 1 9 6 7 ) . The e s t i m a t e o f a p a r t i c u l a r r e g r e s s i o n

coefficient may then be over- or underestimated. The amount of bias in the coefficients depends on the variables that have not been measured and it is difficult to make a judgement on the magnitude of the bias.

However, where the primary objective of the analysis is to provide an accurate prediction of Y rather than separate the effects of the various independent variables, the bias may be of assistance. If the unknown variables were good predictors of Y and have stable

relationships with particular X variables, then the prediction of Y would be improved as the coefficients of the independent variables incorporate the effects of the unknowns.

The REX program [op. cit. , p.34] uses a least squares type of linear regression analysis. Four major assumptions are made in this type of analysis. The assumptions concern the residuals (or errors) present. Therefore, after fitting a curve to the data to form a

mathematical model, it is necessary to ascertain whether the residuals exhibit tendencies which confirm, or at least do not deny, the

assumptions. The assumptions are: 1. Homogeneity of the variance; 2. There is no measurement error;

3. There is no serial correlation between residuals;

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